# Re: [isabelle] Higher-order matching against schematic variables

```On 18/11/10 16:07, Lawrence Paulson wrote:
```
```I can't see the answer to this, but something complicated is going on when you match  (?f::((?'a=>?'b)=>?'c)) ?stuff against x y where x :: nat =>  nat.
```
```
```
Thanks, Larry. Indeed, even without the predicate, it gives the same problem when using that pattern, i.e.
```
lemma lem: "g a = 0"
sorry

vs

lemma lem: "(x::nat=>nat) y = 0"
sorry

Now, if we instead use a simpler pattern:

val trm1 = term_of @{cpat "(?f::?'a=>?'b) ?stuff = ?v"};

matching it against "g a = 0" gives 4 matchers:

1. [?f::nat => nat := %a::nat. a, ?v::nat := 0::nat, ?stuff::nat := g a]

2. [?f::nat => nat := g, ?v::nat := 0::nat, ?stuff::nat := a]

3. [?f::?'a => nat := %b::?'a. g a, ?v::nat := 0::nat,
?stuff::?'a := ?stuff::?'a]

4. [?f::?'a => ?'b := g, ?v::?'b := 0::nat, ?stuff::?'a := a]

But matching it against "(x::nat=>nat) y = 0", gives only 1 matcher:

1. [?f::?'a => ?'b := ?x::nat => nat, ?v::?'b := 0::nat,
?stuff::?'a := ?y::nat]

which is of the same shape of the first lemma's 4th matcher.

```
Perhaps the higher-order matcher somehow fails, since 1. and 3. should be returned by it. 2. is essentially 4., but with the schematic type variables instantiated. Now, is 4. a valid matcher even when ?'a and ?'b can be clearly instantiated (to become 2.)? If it is valid, isn't it redundant to have a separate matcher with the type variables uninstantiated?
```
Thanks
Michael

```
```Larry Paulson

On 18 Nov 2010, at 13:37, Michael Chan wrote:

```
```Hi all,

I have a question with higher-order matching against schematic variables. I can't figure out why the pattern trm1 can't be matched against lem if lem was "A x y -->  x y = 0". However, if the RHS contains constants instead, e.g., "A x y -->  g a = 0", a matcher is found. Since x and y have the same types of g and a respectively, how come there isn't a matcher in the former case? A difference is that x and y are schematic variables, but why can't schematic variables be matched against?

Here's the code:

locale A =
fixes f :: "nat =>  nat"
and a :: nat
assumes ax: "f a = 0"

consts
g :: "nat =>  nat"
a :: nat

lemma lem: "A x y -->  x y = 0"
sorry

ML {*

val trm1 = term_of @{cpat "?P -->  (?f::((?'a=>?'b)=>?'c)) ?stuff = ?v"};
val trm2 = Thm.prop_of @{thm lem};

val mtch_seq = let
val init = Envir.empty 0
val ctxt = @{context}
val (Const ("Trueprop",_) \$ trm2) = Thm.prop_of (ProofContext.get_thm ctxt "lem")
in
Unify.matchers @{theory} [(trm1,trm2)]
end;

val seq as SOME (mtch,_) = nthseq 0 mtch_seq;
pretty_env @{context} (Envir.term_env mtch);
*}

Thanks
Michael

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Telephone Number: +44-131-651-3077,
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```
```

```
```

--
Postal Address: School of Informatics, University of Edinburgh,
Room 2.05, Informatics Forum, 10 Crichton Street,
Edinburgh EH8 9AB, UK.
Telephone Number: +44-131-651-3077,
Fax Number: +44-131-650-6899,
Email: M.Chan at ed.ac.uk
Web Page: http://homepages.inf.ed.ac.uk/mchan/

The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.

```

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